## MSG Algorithm

A polynomial algorithm based on the 5 axioms, algorithm MSG, (maximal structure generation), yields a mathematically rigorous but the simplest super-structure, i.e., the maximal structure. The maximal structure of synthesis problem comprises all the combinatorially feasible structures capable of yielding the specified products from the specified raw materials. Certainly, the optimal network or structure is among these feasible structures. These flowsheets range from the simplest to the most complex, or complete, which is represented by the maximal structure itself. Obviously, the optimal structure in terms of a specific objective function, often the cost, is contained in the maximal structure; nevertheless, the simplest is not necessarily the optimal.

## SSG Algorithm

Algorithm SSG (solution structure generation) renders it possible to generate all the solution structures, i.e., it gives rise to the computational procedure for generating the solution structures. In other words, algorithm SSG unveils every feasible flowsheet of the process of interest. Algorithm SSG generates all the solution structures representing the combinatorially feasible flowsheets from the maximal structure. Moreover, these flowsheets are deemed feasible if they can be optimized in terms of the profit or any other appropriate objective function. Consequently, they can be ranked according to the magnitude of the objective function.

## ABB Algorithm

There are several aspects by which we can set the goal of an optimization problem. So we can talk about executing in shortest time, finding the most reliable process and minimizing/maximizing cos tor profit can also be determinative. In the following section, You can get acquainted with a mixed-integer linear programming model, by which a profit-optimal solution can be determined by an arbitrary P-Graph.

Let M be the set of materials, and M=R ∪I ∪P, where R, I and P are the sets of raw materials, intermediate materials and products. Let O be the set of operating units, and let us also use the following two sets, by which the constraints are easier to define.:

- φ-(m) : such operating units, which are able to produce
*m* - φ+(m): such operating units, which use m on their input

Considering the operating units, the following parameters can be added:

- For ∀o∈O let be a
*c*capacity value,_{o}*fix*fix cost,_{o}- and a
*prop*proportional cost._{o}

- For ∀r∈R let be
- the
*price*(obviously the price of a material)_{r} *max*as an upper bound, which gives the maximum quantity of a raw material._{r}

- the
- For ∀i∈I let be
*price*the selling price_{i}*max*is also an upper bound which controls the remaining quantity in a material node_{i}*penal*represents the value coming from a penalty rate which is considered if a node has remaining quantity not to consumed by any operating unit_{i}

- For ∀p∈P
*price*the income for every product_{p}*min*, is a lower bound how much product has to be produced_{p}*max*can be an upper bound for the selling quantity examining field surveys_{p}

Considering mass balance, the following parameters can be added:

- For ∀o∈O and for ∀m∈M
*input_ratio*is the rate of_{o,m}*m*on the input of*o**output_ratio*is the rate of_{o,m}*m*on the output of*o*

The decision variables of the mathematical model are the following:

- ∀o∈O there is an x_o∈R_0^+ and a y_o∈{0,1} variable. These represent the quantity flowing through the operating unit and the existence of it whether it is considered in the structure or not.

The constraints of the model are the following:

The income, the profit coming from the difference between saling intermediate materials and paying its penalties, and the costs of buying raw materials and using products can be seen in the above expression.